1. Field of the Invention
The present invention relates to novel methods for the one-sided printing and manufacture of a Mxc3x6bius strip. Methods of the invention are particularly well suited for constructing relatively seamless Mxc3x6bius strips for use in a wide variety of retail items.
2. Background
August F. Mxc3x6bius was a mathematician and astronomer in the 18th century who wrote extensively about what has now been named after him, the Mxc3x6bius strip. The Mxc3x6bius strip is a continuous one-sided surface that can be formed from a rectangular strip by rotating one end 180xc2x0 and attaching it to the other end.
The Mxc3x6bius strip has enjoyed a place of prominence in the world of geometry as a curiosity. For example, children often are taught to construct a Mxc3x6bius strip from a piece of paper. A traditional method for creating a Mxc3x6bius strip includes cutting a strip of paper, e.g., about an inch wide and a foot long, putting a half-twist in the strip, then joining the two ends together, e.g., with tape or other adhesive. In that way, one can turn a two-sided strip of paper into a one-sided Mxc3x6bius strip.
Though the child""s paper strip illustrates the concept of the Mxc3x6bius strip, the method used to create it results only in a crude form of a Mxc3x6bius strip. For example, that method produces a strip with a prominent seam which degrades the illusion of a strip having an infinite surface. As such, it does not reflect the true nature of the Mxc3x6bius strip.
The Mxc3x6bius strip concept has been employed to construct various devices. For example, U.S. Pat. No. 4,253,836 reported use of the Mxc3x6bius strip design in the construction of power transmission belts. Other patents have reported use of the Mxc3x6bius strip concept in game puzzles. For example, U.S. Pat. No. 4,384,717 discloses a puzzle comprising two Mxc3x6bius strip loops linked together to form a composite Mxc3x6bius strip. See, also, U.S. Pat. No. 5,324,037, for its report of a game puzzle in the form of a Mxc3x6bius strip comprising multiple columns and rows of block-like pieces.
The Mxc3x6bius strip concept has potential for use in a variety of applications. However, in order to produce a high quality Mxc3x6bius strip, e.g., for use as a retail item, notable challenges are presented in terms of manufacturing. As noted above, it is critical for the final product to be as seamless as possible. Additionally, it often is desirable for the Mxc3x6bius strip to have printed matter on it. In that case, the alignment of the printing of the strip is critical in creating the illusion of a never-ending image. There are no printing machines that can readily adapt to the unique challenges presented in creating a high quality Mxc3x6bius strip in a cost-effective manner.
Thus, it would be desirable to have new methods for manufacturing Mxc3x6bius strips. It would be highly desirable to develop such methods to create a relatively seamless Mxc3x6bius strip which can be printed on at least one side prior to final construction. Such methods would be useful for the construction of Mxc3x6bius strips for use in a variety of retail items, such as puzzles, gift and other novelty items.
I have discovered novel methods for the one-sided printing and manufacture of a Mxc3x6bius strip, which include a pre-assembled Mxc3x6bius strip and both a single-piece puzzle and a multi-piece puzzle. Methods of the invention may be used to create a relatively seamless Mxc3x6bius strip when assembled.
Methods of the invention are particularly suited for the one-sided printing and manufacture of Mxc3x6bius strips for use in a variety of retail items, such as puzzles, gift and other novelty items, and for use as a marketing tool.
In particular, methods of the invention comprise creating a printable rectangular image which may optionally contain color, words or other graphics which are continuous. That is, the left end of the image aligns with the right end of the image to produce a 360xc2x0 image; arranging three substantially identical copies of the image such that one copy is juxtaposed to a second copy and a third copy is centered beneath and adjacent to the first two copies; printing this composite image on a material suitable for use as a Mxc3x6bius strip, preferably having an adhesive backing protected by a removable liner; setting a die layout in a desired configuration; and die-cutting the Mxc3x6bius strip in the desired design. Assembly of the Mxc3x6bius strip generally comprises removing the liner from the adhesive; folding the die-cut piece along a score that runs across the middle of the piece so that the top and bottom halves adhere to one another; rotating one end of the strip about 180xc2x0; and joining the two ends of the strip by aligning the two ends so that they match, thus joining the remaining exposed adhesive to form a twisted closed loop.
While preferred embodiments of the invention comprise use of a printed image, the strip also may be created without graphics, in which case, only the die-cutting and folding methods would apply.
Methods of the invention may be used to produce Mxc3x6bius strip puzzles, e.g., one-piece type puzzles or multi-piece puzzles. Such puzzles may be constructed in many different configurations, and may optionally contain color, or a printed or graphic image as described above.
Other aspects of the invention are disclosed infra.